where zt = (εt)/(sqrt(ht))zt=εtht, E[|zt − i|]E[|zt-i|] denotes the expected value of |zt − i||zt-i| and εt ∣ ψt − 1 = N(0,ht)εt∣ψt-1=N(0,ht) or εt ∣ ψt − 1 = St(df,ht)εt∣ψt-1=St(df,ht). Here StSt is a standardized Student's tt-distribution with dfdf degrees of freedom and variance htht, TT is the number of terms in the sequence, ytyt denotes the endogenous variables, xtxt the exogenous variables, bobo the regression mean, bb the regression coefficients, εtεt the residuals, htht the conditional variance, dfdf the number of degrees of freedom of the Student's tt-distribution, and ψtψt the set of all information up to time tt.

Note: if the yt = μ + εtyt=μ+εt, where μμ is known (not to be estimated by nag_tsa_uni_garch_exp_estim (g13fg)) then (1) can be written as ytμ = εtytμ=εt, where ytμ = yt − μytμ=yt-μ. This corresponds to the case No Regression and No Mean, with ytyt replaced by yt − μyt-μ.

The first element contains the standard error for αoαo and the next iq elements contain the standard errors for
αiαi, for i = 1,2, … ,qi=1,2,…,q. The next iq elements contain the standard errors for
φiϕi, for i = 1,2, … ,qi=1,2,…,q. The next ip elements are the standard errors for
βjβj, for j = 1,2, … ,pj=1,2,…,p.

If dist = 'T'dist='T', the next element contains the standard error for dfdf, the number of degrees of freedom of the Student's tt-distribution.

The first element contains the scores for αoαo, the next iq elements contain the scores for
αiαi, for i = 1,2, … ,qi=1,2,…,q, the next
iq elements contain the scores for φiϕi, for i = 1,2, … ,qi=1,2,…,q, the next ip elements are the scores for
βjβj, for j = 1,2, … ,pj=1,2,…,p.

If dist = 'T'dist='T', the next element contains the scores for dfdf, the number of degrees of freedom of the Student's tt-distribution.